An absorbing game ebbed and flowed from the first whistle. United missed opportunities to forge ahead through Jesse Lingard and Etzaz Hussain, and that profligacy was punished when Milan Lalkovic pipped Sam Johnstone to a loose ball and slotted home.
Having taken the lead against the run of play, Chelsea began to dominate and doubled their lead when Philipp Prosinek capped a fine passing move with an emphatic header.
Within seconds of the restart, however, United halved the arrears when left-back McGinty smashed home an angled effort from just outside the Chelsea area.
The second half followed the same end-to-end pattern as the first, as Bobby Devyne wasted a decent opening for hosts, before Larnell Cole's fierce effort was saved by Blues goalkeeper Jamal Blackman.
Chelsea seemed the likelier to grab the game's fourth goal, as Devyne forced a solid save from Johstone and substitute Mesca curled just over the bar, but United snatched parity with 17 minutes remaining.
Ryan Tunnicliffe's burst into the area created an opening for Cofie and, although the striker's shot was blocked, Keane was on hand to roll home the finish and haul the Reds level.